It is well known that if \(E[p]\)  is irreducible as a Galois module, then \(\mathcal {L}(E,T)\in \Lambda \)  (see [1]}). By the Weierstrass Preparation theorem,
\(\mathcal {L}(E/\mathbb {Q}, T)=p^{\mu }\cdot u(T)g(T)\) 
